Queueing Problems[1]

  1. Customers arrive at a local 7-11 at the rate of = 40 per hour (and follow a Poisson process). The only employee in the store can check them out at a rate of = 60 per hour (following an exponential distribution). Compute the following:
  2. The percentage of time that the employee is busy with the checkouts.
  1. The average length of the queue.
  1. The average number of customers in the system.

  1. The average time spent waiting in the queue.
  1. The average time in the system.
  1. There is only one copying machine in the student lounge of the business school. Students arrive at a rate of  = 40 per hour (according to a Poisson distribution). Copying takes an average of 40 seconds, or  = 90 per hour (according to an exponential distribution). Compute the following:
  2. The percentage of time that the machine is used.
  1. The average length of the queue.

  1. The average number of students in the system.
  1. The average time spent waiting in the queue.
  1. The average time in the system.
  1. Due to a recent increase in business, a law firm secretary must now word-process an average of 20 letters a day (assume a Poisson distribution). It takes him approximately 20 minutes to type each letter (assume an exponential distribution). Assuming the secretary works 8 hours a day:
  1. What is the secretary’s utilization rate?

  1. What is the average waiting time before the secretary word-processes a letter?
  1. What is the average number of letters waiting to be done?
  1. What is the probability that the secretary has more than 5 letters to do?

  1. Sam the Vet is running a rabies-vaccination clinic for dogs at the local grade school. Sam can “shoot” a dog every three minutes. It is estimated that the dogs will arrive independently and randomly throughout the day at a rate of one dog every 6 minutes according to a Poisson distribution. Also assume that Sam’s shooting times are exponentially distributed. Compute the following:
  1. The probability that Sam is idle.
  1. The proportion of time that Sam is busy.
  1. The average number of dogs being vaccinated and waiting to be vaccinated.

  1. The average number of dogs waiting to be vaccinated.
  1. The average time a dog waits before getting vaccinated.
  1. The average amount of time a dog spends waiting in line and being vaccinated.

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Operations ManagementProfs. Juran and Pinedo

[1] These problems appear in Operations Management, 6th Edition, by Jay Heizer and Barry Render (Prentice-Hall, 2001, ISBN 0-536-62524-7) and are used here with permission.